Math 104 Assignment 5 common mistakes

• Several issues on problem 1.C.:
  • Some people went from the statement $a_n \leq c_n \leq b_n$ to the statement $\abs{a_n-L}\leq\abs{c_n-L}\leq\abs{b_n-L}$, which need not be true.
  • Some people assumed that $(c_n)_n$ converged without proving it.
  • Some people used the Squeeze Theorem; this is circular, since this problem is the Squeeze Theorem.
• While it is true that, for example, $\lim_{k\to\infty}d(x, a_k) = d(x, \lim_{k\to\infty} a_k)$, this is not something which can be used without justification.
• On problem 4.A.:
  • Some people assumed that $f$ was bounded on all of $E$, rather than just on $E \cap B_r(p)$.
  • Some people assumed that $\lim_{x\to p}f(x)$ existed, which need not be the case.